Evaluation of Tachiya’s algebraic infinite products involving Fibonacci and Lucas numbers

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Research Article| September 06 2011

Jonathan Sondow

209 West 97th Street ♯6F, New York, NY 10025, USA

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jsondow@alumni.princeton.edu

AIP Conf. Proc. 1385, 97–100 (2011)

https://doi.org/10.1063/1.3630044

In 2007, Tachiya gave necessary and sufficient conditions for the transcendence of certain infinite products involving Fibonacci numbers

F_{k}

and Lucas numbers

L_{k} .

In the present note, we explicitly evaluate two classes of his algebraic examples. Special cases are

\prod_{n = 1}^{\infty} (1+ \frac{1}{F_{2^{n} +1}}) = \frac{3}{φ}, \prod_{n = 1}^{\infty} (1+ \frac{1}{L_{2^{n} +1}}) = 3−φ,

where

φ = (1+ \sqrt{5})/2

is the golden ratio.

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2011

American Institute of Physics

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